Monte Carlo Simula/on and Copula Func/on. by Gerardo Ferrara

Transcription

1 Monte Carlo Simula/on and Copula Func/on by Gerardo Ferrara

2 Introduc)on A Monte Carlo method is a computational algorithm that relies on repeated random sampling to compute its results. In a nutshell, instead of performing long complex calculations, we perform a large number of experiments using a quasi random number generation and see what happens. Monte Carlo methods tend to be used when it is infeasible or impossible to compute an exact result with a deterministic algorithm.

3 Background/History Monte Carlo from the gambling town of the same name (no surprise). First applied in 1947 to model diffusion of neutrons through fissile materials. Limited use because time consuming. Much more common since late 80.

4 The steps in Monte Carlo simulation corresponding to the uncertainty propagation are relatively simple: Step 1: Create a parametric model, y = f(x1, x2,..., xq). Step 2: Generate a set of random inputs, xi1, xi2,..., xiq. Step 3: Evaluate the model and store the results as yi. Step 4: Repeat steps 2 and 3 for i = 1 to n. Step 5: Analyze the results using histograms, summary statistics, confidence intervals, etc.

5 EXAMPLE Area of a figure Cover the figure by a grid, calculate the number of grid cells which are inside and this gives you the area. Shoot at random at the figure. Count the bullets that hit it. The area of then figure is S=(Nhit/Ntotal)*S(rectangle)

6 Monte Carlo Methods A Monte Carlo simulation creates samples from a known distribution. For example, if you know that a coin is weighted so that heads will occur 90% of the time, then you might assign the following values: X 0 1 f X (x)

7 Monte Carlo Methods (cont.) If you tossed the coin, the expected value would be 0.9 However, a sample simulation might yield the results 1, 1, 1, 0, 1, 1, 0, 1, 0, 1. The average of the sample is 0.7 (close, but not the same as the expected average).

8 Value at Risk (VaR) We are X percent certain that we will not lose more than V dollars in time T. Function of confidence level X and time T.

9 Pseudo Random Number Generators Monte Carlo simulations are based on computer generation of pseudo random numbers. Starting point is generation of sequence of independent, identically distributed uniform (U(0,1)) random variables: U(0,1) random numbers of direct interest in some applications; More commonly, U(0,1) numbers transformed to random numbers having other distributions. 9

10 Example Use of Simulation: Monte Carlo Integration Common problem is estimation of ( x ) dx where f is a Ω function, x is vector and Ω is domain of integration Monte Carlo integration popular for complex f and/or Ω. b Special case: Estimate f ( x ) dx for scalar x, and limits of a integration a, b. One approach: f Let p(u) denote uniform density function over [a, b] Let U i denote i th uniform random variable generated by Monte Carlo according to the density p(u) Then, for large n: b a b a f ( x) dx f ( U ) n n i= 1 i 10

11 Numerical Example of Monte Carlo Integration b Suppose interested in sin( xdx ) 0 Simple problem with known solution. Considerable variability in quality of solution for varying b Accuracy of numerical integration sensitive to integrand and domain of integration. Integral estimates for varying n n = 20 n = 200 n = 2000 b = π (ans.=2) b = 2π (ans.=0)

12 Homework Exercise 1 This problem uses the Monte Carlo integration technique to estimate for varying a, b, and n. Specifically: b a 2x 2 + 3x 1dx (a) To at least 3 post-decimal digits of accuracy, what is the true integral value when a = 0, b = 1? And for a = 0, b = 4? (b) Using n = 20, 200, and 2000, estimate (via Monte Carlo) the integral for the two combinations of a and b in part (a). (c) Comment on the relative accuracy of the two settings. 12

13 Copulas Suppose you want to generate samples from some distribution with probability density f(x). All you need is a source of uniform random variables, because you can transform these random variables to have the distribution that you want (Sklar s Theorem).

14 General algorithm Generate (w1,w2) from a Multivariate Normal. Get u = F(w1), v = F(w2) where F(x) is normal cumulative distribution function (CDF). Generate x = G -1 (u), y = G -1 (v) where G -1 is empirical CDF from data. The distribution multivariate normal distribution is important; this is what controls dependence at the uniform density stage.

15 What is an empirical CDF? Given a vector S t of observations (then you can use the ecdf function in R). The methodology assigns a 1/n probability to each observation, orders the data from smallest to largest in value, and calculates the sum of the assigned probabilities up to and including each observation. The result is a step function that increases by at each datum. p = G(z) = fraction(s t z) G -1 (p) = quantile(s t, p)